Over the past decade, the utilization of digital data signaling systems has expanded at a tremendous rate due to the increased use of digital equipment for producing such digital data. Such digital signaling systems have thus been developed to handle signals having data rates anywhere from very low speed systems, such as teletypewriters which generally do not exceed 100 bits per second, to very high speed systems such as PCM color TV systems which can operate at 100 megabits per second or higher.
Although a great deal of work has thus been done in developing such digital data signaling systems, a persistent problem has been how to best deal with the effects of intersymbol interference (ISI). This problem arises on channels in which the pulses representing different transmitted symbols overlap in time to some degree. The overlap of the pulses causes a greater difficulty in the reliable estimation of the transmitted symbol at the receiver. For example, the output of a channel which creates intersymbol interference is formally represented as: ##EQU1## where
x.sub.i =Channel input pulse at time i
f.sub.o =Largest sample of the channel input response
f.sub.k =Channel coefficient which is the fraction of the (i-k).sup.th input pulse that contributes to the channel output at the i.sup.th symbol time, i.e., intersymbol interference
y.sub.i =Channel output at the i.sup.th symbol time
From Equation 1, one observes that the channel output is, in general, influenced by other input symbols which surround the present input pulse, x.sub.i. If the channel had no intersymbol interference, f.sub.o =1 and f.sub.k =0, k.noteq.0. In this case, the channel output, y.sub.i, is proportional to the input, x.sub.i, with no intersymbol interference from surrounding input symbols.
The intersymbol interference phenomenon is often graphically displayed by a plot of the "eye pattern." This eye pattern is simply a plot of all possible values for y.sub.i obtained by assuming all possible sequences of input symbols x.sub.i. For example, assuming binary input symbols x.sub.i =.+-.1 and x.sub.i-1 =.+-.1 and a channel characterized by f.sub.o =1 and f.sub.1 =0.25, with all other f.sub.k =0, one obtains from Equation 1 the outputs y.sub.i =1.25, 0.75, -0.75 and -1.25 for all four combinations of {x.sub.i, x.sub.i-1 }. These values are plotted in FIG. 1. In the absence of intersymbol interference (ISI), the values for channel output of .+-.1 as shown by the dashed lines in FIG. 1 would be observed. With ISI, it can be seen that the ideal outputs are modified to either .+-.1.25 or .+-.0.75 depending upon the data pair, {x.sub.i, x.sub.i-1 }. The minimum spacing, called the "eye opening," between data levels has been reduced from 2.0 for no ISI, to 1.5 with ISI. With added noise, the probability of error is increased due to the ISI since less margin exists to a slicing threshold at zero.
Conceptually, equalizers recover some of the performance loss brought about by ISI by attempting to open the eye pattern at the receiver. Usually a penalty is incurred in opening the eye pattern through noise enhancement. That is, when the eye pattern is opened by the equalizer, more noise exists on the equalizer output sample than on the channel output sample. For one type of theoretically perfect equalizer, this noise enhancement phenomenon can be displayed in the following way. Using the delay operation, D, sequences of channel outputs y.sub.i can be represented formally as coefficients in the polynomial EQU y(D)=y.sub.o +y.sub.1 D+y.sub.2 D.sup.2 . . . (2)
and channel inputs by EQU x(D)=x.sub.o +x.sub.1 D+x.sub.2 D.sup.2 . . . (3)
With this characterization, and Equation (1), we can write EQU y(D)=x(D)f(D) (4)
where EQU f(D)=.SIGMA.f.sub.k D.sup.k ( 5)
Thus, from Equation (4), the action of the channel is seen to be a polynomial multiplication of the input polynomial, x(D), by the channel polynomial, f(D). Obviously, the effect of the channel can be completely undone by dividing the reveived sequence, y(D), by the polynomial, f(D). This division can be accomplished by feeding the received sequence into a recursive filter. Equivalently, the equalization can be performed by feeding the received sequence into an infinite length transversal filter characterized by tap gain polynomial EQU t(D)=1/f(D) (6)
Considering the specific f(D) polynomial producing the eye pattern of FIG. 1, i.e. f(D)=1+0.25D, the equalizer is defined by ##EQU2##
Equation (7) characterizes the infinite transversal filter which is required to perfectly equalize the channel. The coefficients of t(D) specify the tap gains for the equalizer. Of course, as a practical matter the transversal equalizer would have to be truncated at some finite length, say n, where (0.25).sup.n is insignificant.
Turning now to the question of noise enhancement, if the infinite length transversal filter defined in Equation (7) is used to perfectly undo the ISI created by the channel, the noise enhancement factor is given by the sum of the squares of the coefficients of the t(D) polynomial, i.e.: ##EQU3## The quantity, N.sub.e, gives the factor by which the variance of the equalizer input sample noise is increased at the output of the equalizer. Thus, one pays a penalty for the privilege of equalizing the channel and this penalty is usually in the form of noise enhancement. The improved performance may, however, make the penalty very worthwhile. In the above example, the noise is only increased by a factor of 1.067, so the greatly improved eye pattern opening makes such a penalty very worthwhile.
The foregoing background has been given to show fundamentally how linear equalizers accomplish their function (opening the eye pattern) and the penalty (noise enhancement) incurred in the performance of this function. Although the equalizer examined above was the infinite length transversal filter required to provide zero intersymbol interference, other types of equalizers of finite length exist and algorithms are available for automatically adjusting the tap gains to accomplish equalization. FIG. 2 shows an example of such a finite length linear equalizer structure.
Referring to FIG. 2, the channel output signal, which includes ISI, is fed to an N-stage delay line 10 through a baud rate sampler 12. The outputs of the N-stage delay line 10 are each coupled to a multiplier 14. Each of these multipliers 14 is fed with a weighting function w.sub.i where i=1 . . . N. The weighted outputs from these multipliers 14 are then combined to form a composite equalized output by an adder 16.
With regard to the linear equalizer of FIG. 2, ##EQU4## defines the equalizer transfer function. The linear tap weights, {w.sub.i }, are determined adaptively by one of several known techniques using well-known circuitry not shown in FIG. 2.
As shown in FIG. 2, the output from a linear equalizer is given by a linear weighted combination of tap outputs. Typically, a known training sequence is initially transmitted over the channel and the tap gain adjustment circuitry (not shown) learns appropriate tap gains to use in order to effect equalization of the ISI existing on the channel.
Although the basic linear equalizer shown in FIG. 2 can significantly reduce ISI without substantial degradation of the noise enhancement factor, it suffers from a number of problems. In particular, the requirement of using multipliers for the equalization is relatively time consuming. This can result in limiting the response speed of the receiver. Such limitations can be especially troublesome in situations where high bit rates must be handled.
Another shortcoming of the basic linear equalizer of FIG. 2 is the fact that it does not perform nonlinear equalization. In some cases, the ability to perform nonlinear tap gain weighting may be a more effective equalization technique than linear weighting. Such nonlinear weighting would be useful, for example, in combating nonlinearities occurring in the channel. Even in the absence of such nonlinearities in the channel, nonlinear equalizer can have merit. For example, a nonlinear output versus input characteristic can display a soft limiter function which essentially makes a decision on the selected tap position bit and weights this decision. Such a limiter characteristic can have the beneficial effect of suppressing noise on the equalizer output.
A discussion of the desirability of nonlinear equalizers with regard to linear equalizers in terms of the eye pattern and error probability is set forth in an article by Gottfried Ungerboeck published in the December 1971 issue of the IEEE Transactions on Communication Technology, entitled "Nonlinear Equalization of Binary Signals in Gaussian Noise." In this paper, a very complex nonlinear equalizer is first described. However, since this equalizer is not feasible for actual implementation in a practical system, sub-optimal approximations for nonlinear equalizers are subsequently discussed. To accomplish this, a plurality of nonlinear decision elements can be used which take the form of soft limiters rather than linear tap gains. Although Ungerboeck then proceeds to show that even these sub-optimal nonlinear equalizers are superior to an optimal linear equalizer, the system which he proposes is still relatively slow in operation. Also, although Ungerboeck mentions the possibility of making the system adaptive, no example is given for this, and Ungerboeck specifically indicates that further experimentation would be necessary in this ragard.